OFFSET
3,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 3..1000
FORMULA
-(n+6)*(n-3)*a(n) +2*(3*n^2+3*n-20)*a(n-1) +(-9*n^2+15*n+20)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Jun 20 2013
From G. C. Greubel, Mar 19 2021: (Start)
G.f.: (1-x)*((1-3*x)*(1 -6*x +9*x^2 -3*x^3) -(1-x)*(1 -6*x +9*x^2 -x^3)*sqrt(1-4*x))/(2*x^6).
G.f.: (1-x)*x^3*C(x)^9, where C(x) is the g.f. of the Catalan numbers (A000108).
E.g.f.: exp(2*x)*(BesselI(3, 2*x) - BesselI(6, 2*x)).
a(n) = binomial(2*n, n-3) - binomial(2*n, n-6) = A026009(2*n, n-3).
a(n) = f(n) - f(n-1), where f(n) = Sum_{j=0..n-3} C(n-j-3)*(C(j+7) -6*C(j+6) +10*C(j+5) -4*C(j+4)) and C(n) are the Catalan numbers. (End)
From G. C. Greubel, Mar 22 2021: (Start)
a(n) = C(n+5) -8*C(n+4) +22*C(n+3) -25*C(n+2) +11*C(n+1) -C(n).
a(n) = (9/20)*(binomial(n,3)/binomial(n+6,5))*(3*n^2 +3*n +20)*C(n). (End)
MATHEMATICA
Table[Binomial[2*n, n-3] - Binomial[2*n, n-6], {n, 3, 30}] (* G. C. Greubel, Mar 19 2021 *)
PROG
(Sage) [binomial(2*n, n-3) - binomial(2*n, n-6) for n in (3..30)] # G. C. Greubel, Mar 19 2021
(Magma) [Binomial(2*n, n-3) - Binomial(2*n, n-6): n in [3..30]]; // G. C. Greubel, Mar 19 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved